Extended Fuller index, sky catastrophes and the Seifert conjecture

Abstract: We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields $X$ on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at $X$ has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on $S ^{2k+1} $. A sky catastrophe, is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller's foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.